A set is an unordered collection of objects.


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1 Section 2.1
2 Sets A set is an unordered collection of objects. the students in this class the chairs in this room The objects in a set are called the elements, or members of the set. A set is said to contain its elements. The notation a A denotes that a is an element of the set A. If a is not a member of A, write a A
3 Describing a Set: Roster Method S = {a,b,c,d} Order not important S = {a,b,c,d} = {b,c,a,d} Each distinct object is either a member or not; listing more than once does not change the set. S = {a,b,c,d} = {a,b,c,b,c,d} Elipses ( ) may be used to describe a set without listing all of the members when the pattern is clear. S = {a,b,c,d,,z }
4 Roster Method Set of all vowels in the English alphabet: V = {a,e,i,o,u} Set of all odd positive integers less than 10: O = {1,3,5,7,9} Set of all positive integers less than 100: S = {1,2,3,..,99} Set of all integers less than 0: S = {., 3,2,1}
5 Some Important Sets N = natural numbers = {0,1,2,3.} Z = integers = {,3,2,1,0,1,2,3, } Z + = positive integers = {1,2,3,..} R = set of real numbers R + = set of positive real numbers C = set of complex numbers. Q = set of rational numbers
6 SetBuilder Notation Specify the property or properties that all members must satisfy: S = {x x is a positive integer less than 100} O = {x x is an odd positive integer less than 10} O = {x Z + x is odd and x < 10} A predicate may be used: S = {x P(x)} Example: S = {x Prime(x)} Positive rational numbers: Q= {x R x = p/q, for some positive integers p,q}
7 Interval Notation [a,b] = {x a x b} [a,b) = {x a x < b} (a,b] = {x a < x b} (a,b) = {x a < x < b} closed interval [a,b] open interval (a,b)
8 Universal Set and Empty Set The universal set U is the set containing everything currently under consideration. Sometimes implicit Sometimes explicitly stated. Contents depend on the context. The empty set is the set with no elements. Symbolized, but {} also used.
9 Russell s Paradox Let S be the set of all sets which are not members of themselves. A paradox results from trying to answer the question Is S a member of itself? Related Paradox: Henry is a barber who shaves all people who do not shave themselves. A paradox results from trying to answer the question Does Henry shave himself?
10 Some things to remember Sets can be elements of sets. {{1,2,3},a, {b,c}} {N, Z, Q, R} The empty set is different from a set containing the empty set. { }
11 Set Equality Definition: Two sets are equal if and only if they have the same elements. Therefore if A and B are sets, then A and B are equal if and only if. We write A = B if A and B are equal sets. {1,3,5} = {3, 5, 1} {1,5,5,5,3,3,1} = {1,3,5}
12 Venn Diagrams shows all possible logical relations between a finite collection of sets U U A B A B C
13 Subsets Definition: The set A is a subset of B, if and only if every element of A is also an element of B. The notation A B is used to indicate that A is a subset of the set B. A B holds if and only if is true. 1. Because a is always false, S,for every set S. 2. Because a S a S, S S, for every set S.
14 Showing a Set is or is not a Subset of Another Set Showing that A is a Subset of B: To show that A B, show that if x belongs to A, then x also belongs to B. Showing that A is not a Subset of B: To show that A is not a subset of B, A B, find an element x A with x B. (Such an x is a counterexample to the claim that x A implies x B.) Examples: 1. The set of all computer science majors at your school is a subset of all students at your school. 2. The set of integers with squares less than 100 is not a subset of the set of nonnegative integers.
15 Another look at Equality of Sets Recall that two sets A and B are equal, denoted by A = B, if and only if Using logical equivalences we have that A = B if and only if This is equivalent to A B and B A
16 Proper Subsets Definition: If A B, but A B, then we say A is a proper subset of B, denoted by A B. If A B, then is true. Venn Diagram B A U
17 Set Cardinality Definition: If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is finite. Otherwise it is infinite. Definition: The cardinality of a finite set A, denoted by A, is the number of (distinct) elements of A. Examples: 1. ø = 0 2. Let S be the letters of the English alphabet. Then S = {1,2,3} = 3 4. {ø} = 1 5. The set of integers is infinite.
18 Power Sets Definition: The set of all subsets of a set A, denoted P(A), is called the power set of A. Example: If A = {a,b} then P(A) = {ø, {a},{b},{a,b}} If a set has n elements, then the cardinality of the power set is 2ⁿ. (In Chapters 5 and 6, we will discuss different ways to show this.)
19 Tuples The ordered ntuple (a 1,a 2,..,a n ) is the ordered collection that has a 1 as its first element and a 2 as its second element and so on until a n as its last element. Two ntuples are equal if and only if their corresponding elements are equal. 2tuples are called ordered pairs. The ordered pairs (a,b) and (c,d) are equal if and only if a = c and b = d.
20 Cartesian Product Definition: The Cartesian Product of two sets A and B, denoted by A B is the set of ordered pairs (a,b) where a A and b B. Example: A = {a,b} B = {1,2,3} A B = {(a,1),(a,2),(a,3), (b,1),(b,2),(b,3)} Definition: A subset R of the Cartesian product A B is called a relation from the set A to the set B. (Relations will be covered in depth in Chapter 9. )
21 Cartesian Product Definition: The cartesian products of the sets A 1,A 2,,A n, denoted by A 1 A 2 A n, is the set of ordered ntuples (a 1,a 2,,a n ) where a i belongs to A i for i = 1, n. Example: What is A B C where A = {0,1}, B = {1,2} and C = {0,1,2} Solution: A B C = {(0,1,0), (0,1,1), (0,1,2),(0,2,0), (0,2,1), (0,2,2),(1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,1,2)}
22 Truth Sets of Quantifiers Given a predicate P and a domain D, we define the truth set of P to be the set of elements in D for which P(x) is true. The truth set of P(x) is denoted by Example: The truth set of P(x) where the domain is the integers and P(x) is x = 1 is the set {1,1}
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